Surface plastic flow in polishing of rough surfaces

We present experimental evidence for a new mechanism for how smooth surfaces emerge during repetitive sliding contacts, as in polishing. Electron microscopy observations of Ti-6Al-4V surface with a spherical asperity structure—realized via additive manufacturing—during successive polishing stages suggest that asperity-abrasive contacts exhibit viscous behavior, where the asperity material flows in the form of thin (1–10 μm) fluid-like layers. Subsequent bridging of these layers among neighboring asperities results in progressive surface smoothening. Using analytical asperity-abrasive contact temperature modeling and microstructural characterization, we show that the sliding contacts encounter flash temperatures of the order of 700–900 K which is in the range of the dynamic recrystallization temperature of the material considered, thus supporting the experimental observations. Besides providing a new perspective on the long-held mechanism of polishing, our observations provide a novel approach based on graph theory to quantitatively characterize the evolution of surface morphology. Results suggest that the graph representation offers a more efficient measure to characterize the surface morphology emerging at various stages of polishing. The research findings and observations are of broad relevance to the understanding of plastic flow behavior of sliding contacts ubiquitous in materials processing, tribology, and natural geological processes as well as present unique opportunities to tailor the microstructures by controlling the thermomechanics of the asperity-abrasive contacts.


S1: Calculation of flash temperatures at the asperity-abrasive contacts
FIG. S1. Schematic showing contact (solid line) between the workpiece surface consisting of spherical asperities and the polishing pad at a distance S z (average asperity heights) from the workpiece reference plane (dotted line). Here, the asperity height, z, is measured with respect to the workpiece reference plane.
For a given asperity height (z) distribution, only the asperities for which z > S z and z ≤ S z + 2R are involved in the polishing process, as schematically shown in Fig. S1. Here, the asperity height, z, is measured with respect to the workpiece reference plane (dotted line FIG. S2. Moving circular heat source model for the contact between asperity and abrasive to calculate the temperature rise during polishing. Here, the abrasive is considered as the semiinfinite moving heat source and the asperity acts as a stationary heat source. in the schematic in Fig. S1). We assume that the clearance between the workpiece reference plane and the polishing pad (solid line) is equal to the average surface asperity heights, S z , of the workpiece. The diameter of asperity-abrasive contact (2a) can then be calculated for a given value of S z , asperity radius (R) and height (z) distribution.
Given the radius of contact, we calculate flash temperature by treating the contact as a moving circular heat source (Fig. S2). The heat source intensity is taken as the heat dissipation due to plastic shearing of the metal asperity at the sliding interface. The heat partition between the asperity and the abrasive particle is determined by setting equal the maximum (quasi-steady state) temperatures of the asperity and abrasive particle within the contact, according to Blok's postulate [S1]. Here, we treat the abrasive as a semi-infinite moving body (with velocity V ) over which a stationary heat source (with uniform heat flux) acts. The steady state flash temperature occurring at the contact center can accordingly be given by the first order approximation to Jaegar's circular moving heat source model [S2, S3] as: where, Peclet number, P e2 = V a/2K 2 and K 2 = k 2 /ρ 2 C 2 ≈ 4 × 10 −5 m 2 /s. For V = 5 m/s and contact radius a, we have P e2 = 6.25 × 10 5 a. For the asperity (which is treated as a stationary source), we have: Assuming adiabatic conditions, where plastic dissipation at the interface is completely converted into heat, the total heat flux, q, at the circular contact is given by: By equating the maximum temperatures at the asperity and abrasive surface, we have: We solve for ∆T max for Ti-6Al-4V using the values in Table 1, and the corresponding flash temperature map as a function of asperity height and abrasive-asperity contact radius is shown in Fig. S3(a). Any asperity for which z < S z or z ≥ S z + 2R would not be involved in the polishing process as it would either make no contact with the abrasive or lie outside the asperity-abrasive contact region (solid line in Fig. S1). These two cases are marked as "p" and "q" in Fig. S3(a). Elsewhere, we notice that larger values of R and z result in higher flash temperatures.
While the assumption of abrasive as a semi-infinite plane maybe reasonable during the initial stages of polishing, the configuration is reversed as polishing process progresses. During the intermediate and final stages, polishing maybe represented as individual abrasive particles sliding across a semi-infinite workpiece surface. For this latter configuration, we assume abrasive particles as sliding conical indenters plastically deforming the workpiece surface. Again for this case, the problem is that of a moving semi-infinite body (workpiece surface) over which stationary heat source (abrasive-workpiece surface contact) acts. The maximum flash temperature rise at the contact in this case is given as: The calculated sliding temperatures for this configuration are slightly larger than those in the earlier configuration where abrasive was taken as a semi-infinite plane (Fig. S2). The difference between temperature estimates for these two configurations is within 20% (at a contact radius of ∼40 µm) for the contact areas considered here. In both the configurations, for ∼ 30% of the sliding contacts, maximum flash temperatures are above the dynamic recrystallization temperature of the alloy (∼ 700 K).
Similar calculations for Ta 2 O 5 showed the flash temperature to be in the range of 750 K.
In this case, the average radius of the asperity-abrasive contact area was inferred from

S2: Graph representation of topological evolution
The phenomena of bridging of the neighboring asperities is analyzed as an evolving random planar graph G(t) = (V, E(t)) where the nodes V represent the asperities and the edge weights E(t) denote the propensity of a pair of neighboring nodes to bridge evolving over time. As observed from the in situ electron micrographs (see Fig. S5), the evolving morphological features, here the particle morphology, are embedded in a two-dimensional plane. To ensure that the graph-based model is consistent with the planar disposition of the surface features, we subscribe to the following planar graph representation in this study: Given the neighborhood structure N i (t) of node i, the propensity e ij (t) ∈ E(t) of a node i to bridge with its n th nearest node j ∈ N i (t) can be expressed as:

Definition
Here, P ( b ij (t)) denotes the prior, i.e., the probability of bridging before the neighborhood structure, N i (t), ∀i ∈ V is gathered from the SEM image sequence. In the absence of information on N i (t), a uniform distribution is assigned to P ( b ij (t)) as an initial (non-informative) prior. To capture the neighborhood information N i (t) from these images, we note that the asperities on the sample surface (as estimated from SEM images, see Fig. S5) follow a homogeneous Poisson distribution. From the distribution theory, we can show that the square of inter-asperity distance ρ 2 ) of an asperity i to its n th nearest neighboring asperity j follows a chi-square distribution with 2n degrees of freedom. Therefore, we have: where ξ ij (t) = 2πλρ 2 ij (t), λ is the asperity surface density and ρ ij (t) = ||v i − v j || 2 − (r i (t) + r j (t)).
From the in situ SEM image sequence shown in Fig. S5(a), we notice that as the pol-ishing process ensues, asperities progressively bridge and is reflected by an increase in the connectivity of the representative asperity network (Fig. S5(b)). To quantify this evolution pattern (we show in the sequel that this quantification would also serve to validate the planar random graph model), we track the second smallest eigenvalue λ 2 (also called the Fiedler value) of the graph Laplacian, L(t) representing the degree of each node and is given as: It has been established that the second largest eigenvalue of L(t) captures the algebraic connectivity in the graph, also called the Fiedler number (λ 2 ) [S4, S5].
The lower bound on λ 2 is calculated using the geometric embedding of planar graph on a unit sphere as presented in [S6], where each of the nodes are represented by non-overlapping semi-spherical caps of radius r i , i ∈ V . For the micrograph in Fig. S5, |V | = 121. A strongly connected network of asperities can be assumed as an ideal close packing of uniform spheres such that each node is connected to at most 6 nearest neighbors. Under such conditions it can be shown that 0.16 ≤ λ 2 ≤ 0.39 holds. The initial value of λ 2 = 0.0203 (see Fig. S5, bottom row) indicates that the degree of each node is < 1. After 450 s of polishing, λ 2 increases to 0.1526 suggesting a minimum degree of 6 among all neighboring asperities. The network structure along with the corresponding λ 2 values is summarized in Fig. S5. Additionally, the linear increase in the value of λ 2 suggests that there are significant topological changes in the surface even during the final stages of polishing process which otherwise are not reflected in the S a or S v measurements (see Fig. 7 in the main text).